Test of Mathematics for University Admission (TMUA) Number, Algebra and Surds Syllabus

Number, Algebra and Surds syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Number, Algebra and Surds in Test of Mathematics for University Admission (TMUA), not a summary of it.

1. 1

   Indices, Surds and Standard Form

   4 topics

   • Laws of Indices
     • Multiplication, division and power rules
     • Negative and zero indices
     • Fractional indices and roots
   • Manipulation of Surds
     • Simplifying surd expressions
     • Rationalising the denominator
     • Surds in solutions to equations
   • Standard Form and Estimation
     • Operations in standard form
     • Order-of-magnitude estimation
   • Rational and Irrational Numbers
     • Recurring decimals as fractions
     • Proving irrationality of surds
2. 2

   Algebraic Manipulation

   4 topics

   • Expanding and Factorising
     • Products of binomials and brackets
     • Common factors and grouping
     • Difference of two squares
   • Algebraic Fractions
     • Simplifying compound fractions
     • Addition, subtraction and division
   • Partial Fractions
     • Distinct linear denominators
     • Repeated and quadratic factors
   • Completing the Square
     • General form a(x+p)^2+q
     • Identifying turning points
3. 3

   Polynomials

   3 topics

   • Polynomial Arithmetic
     • Addition, multiplication and division
     • Long division of polynomials
   • Factor and Remainder Theorems
     • Testing for factors
     • Finding remainders without division
   • Roots of Polynomials
     • Relating roots to coefficients
     • Sketching from factorised form

Number, Algebra and Surds flashcards for Test of Mathematics for University Admission (TMUA)

23 of 51 cards from the Number, Algebra and Surds deck — real questions with worked answers.

1. State the multiplication and division laws of indices for $a^{m}$ and $a^{n}$.

   $a^{m} \times a^{n} = a^{m+n}$ and $\dfrac{a^{m}}{a^{n}} = a^{m-n}$.

2. Simplify the power-of-a-power: what is $\left(a^{m}\right)^{n}$?

   $\left(a^{m}\right)^{n} = a^{mn}$ (multiply the indices).

3. What is the value of $a^{0}$ for any nonzero $a$, and why?

   $a^{0} = 1$. It follows from $\dfrac{a^{m}}{a^{m}} = a^{m-m} = a^{0} = 1$.

4. Express a negative index $a^{-n}$ as a positive-index expression.

   $a^{-n} = \dfrac{1}{a^{n}}$ (the reciprocal of $a^{n}$).

5. What does a fractional index $a^{m/n}$ mean in terms of roots and powers?

   $a^{m/n} = \sqrt[n]{a^{m}} = \left(\sqrt[n]{a}\right)^{m}$.

6. How do you distribute a power over a product and a quotient, e.g. $(ab)^{n}$ and $\left(\tfrac{a}{b}\right)^{n}$?

   $(ab)^{n} = a^{n}b^{n}$ and $\left(\dfrac{a}{b}\right)^{n} = \dfrac{a^{n}}{b^{n}}$.

7. Evaluate $27^{-2/3}$.

   $27^{-2/3} = \dfrac{1}{27^{2/3}} = \dfrac{1}{\left(\sqrt[3]{27}\right)^{2}} = \dfrac{1}{3^{2}} = \dfrac{1}{9}$.

8. What is a surd, and what makes an expression like $\sqrt{2}$ a surd?

   A surd is a root of a positive integer that is irrational, i.e. cannot be simplified to a rational number. $\sqrt{2}$ is a surd because $2$ is not a perfect square.

9. State the two key product/quotient rules for simplifying surds.

   $\sqrt{a}\,\sqrt{b} = \sqrt{ab}$ and $\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$ (for $a,b \geq 0$, $b\neq 0$).

10. Simplify $\sqrt{72}$ to the form $a\sqrt{b}$.

    $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$.

11. How do you rationalise the denominator of $\dfrac{1}{\sqrt{a}}$?

    Multiply numerator and denominator by $\sqrt{a}$: $\dfrac{1}{\sqrt{a}} = \dfrac{\sqrt{a}}{a}$.

12. By what do you multiply to rationalise a denominator of the form $a + b\sqrt{c}$?

    By its conjugate $a - b\sqrt{c}$, using $(a+b\sqrt{c})(a-b\sqrt{c}) = a^{2} - b^{2}c$ (the difference of two squares).

13. Rationalise $\dfrac{1}{3+\sqrt{2}}$.

    $\dfrac{1}{3+\sqrt{2}} \times \dfrac{3-\sqrt{2}}{3-\sqrt{2}} = \dfrac{3-\sqrt{2}}{9-2} = \dfrac{3-\sqrt{2}}{7}$.

14. Why is $\sqrt{a}+\sqrt{b}$ not generally equal to $\sqrt{a+b}$? Give a numerical check.

    They are unequal in general: $\sqrt{9}+\sqrt{16} = 3+4 = 7$, but $\sqrt{9+16}=\sqrt{25}=5$. The square root does not distribute over addition.

15. What is the defining form of standard form (scientific notation)?

    $A \times 10^{n}$ where $1 \leq A < 10$ and $n$ is an integer.

16. Write $0.000\,42$ in standard form.

    $4.2 \times 10^{-4}$.

17. When multiplying numbers in standard form, what happens to the $A$ values and the powers of ten?

    Multiply the $A$ values and add the indices: $(A\times10^{m})(B\times10^{n}) = AB \times 10^{m+n}$, then adjust so $1 \leq A_{\text{new}} < 10$.

18. In estimation, how do you round to one significant figure, and why is it useful?

    Keep only the first nonzero digit (rounding the rest), e.g. $487 \approx 500$. It gives a quick approximate value to sanity-check a calculation.

19. Estimate $\dfrac{38.6 \times 21.4}{9.7}$ using one-significant-figure rounding.

    $\dfrac{40 \times 20}{10} = \dfrac{800}{10} = 80$.

20. Define a rational number.

    A number that can be written as $\dfrac{p}{q}$ where $p,q$ are integers and $q \neq 0$. Its decimal expansion terminates or recurs.

21. Define an irrational number and give two examples.

    A number that cannot be written as a ratio of two integers; its decimal expansion is non-terminating and non-recurring. Examples: $\pi$ and $\sqrt{2}$.

22. Is the sum of a rational number and an irrational number rational or irrational? Justify briefly.

    Irrational. If $r$ (rational) $+ x$ (irrational) were rational, then $x = (\text{rational}) - r$ would be rational, a contradiction.

23. Outline the classic proof that $\sqrt{2}$ is irrational.

    Assume $\sqrt{2}=\dfrac{p}{q}$ in lowest terms. Then $p^{2}=2q^{2}$, so $p$ is even, $p=2k$, giving $q^{2}=2k^{2}$, so $q$ is even too — contradicting lowest terms. Hence $\sqrt{2}$ is irrational.

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